A Computational Study of a Data Assimilation Algorithm for the Two

A Computational Study of a Data
Assimilation Algorithm for the
Two-dimensional Navier–Stokes Equations
Masakazu Gesho∗
Eric Olson†
Edriss S. Titi‡§
May 3, 2015
Abstract
We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control
theory, in the context of the two-dimensional incompressible Navier–
Stokes equations. Our model problem is to recover an unknown reference solution, asymptotically in time, by using continuous-in-time
coarse-mesh nodal-point observational measurements of the velocity
field of this reference solution (subsampling), as might be measured
by an array of weather vane anemometers. Our calculations show
that the required nodal observation density is remarkably less that
what is suggested by the analytical study; and is in fact comparable
to the number of numerically determining Fourier modes, which was
reported in an earlier computational study by the authors. Thus, this
method is computationally efficient and performs far better than the
analytical estimates suggest.
Keywords: Continuous data assimilation; determining nodes; signal synchronization; two-dimensional Navier–Stokes equations; downscaling.
AMS Classification: 35Q30; 93C20; 37C50; 76B75; 34D06.
∗
Department of Mathematics, University of Wyoming, 1000 E. University Ave, Dept.
3036, Laramie, WY 82071, USA. email: [email protected]
†
Department of Mathematics and Statistics, University of Nevada, Reno, NV 89557,
USA. email: [email protected]
‡
Department of Mathematics, Texas A&M University, 3368–TAMU, College Station,
TX 77843, USA. email: [email protected]
§
The Department of Computer Science and Applied Mathematics, Weizmann Institute
of Science, Rehovot 76100, Israel. email: [email protected]
1
1
Introduction
The goal of data assimilation is to provide a more accurate representation
of the current state of a dynamical system by combining observational data
with model dynamics. This allows the influences of new data to be incorporated into a numeric computation over time. Data assimilation is widely
used in the climate sciences, including weather forecasting, environmental
forecasting and hydrological forecasting. Additional information and historical background may be found in Kalnay [13] and references therein.
In 1969 Charney, Halem and Jastrow [5] proposed a method of continuous
data assimilation in which observational measurements are directly inserted
into the mathematical model as it is being integrated in time. To fix ideas,
let us suppose that the evolution of u is governed by the dynamical system
du
= F(u),
dt
u(t0 ) = u0
(1)
and the observations of u are given by the time series p(t) = P u(t) for
t ∈ [t0 , t∗ ], where P is an orthogonal projection onto the low modes. In this
context, the method proposed in [5] for approximating u from the observational data is to solve for the high modes
dq
= (I − P )F(q + p),
dt
q(t0 ) = q0
(2)
where q0 is an arbitrarily chosen initial condition and q + p represents the
resulting approximation of u. Note that if q0 = (I − P )u0 then p + q = u for
all time; however, data assimilation is applied when u0 is not known.
Algorithm (2) was studied by Olson and Titi in [17] and [18] for the
two-dimensional incompressible Navier–Stokes equations


 ∂u − ν∆u + (u · ∇)u + ∇p = f
∂t


(3)
∇·u=0
on the domain Ω = [0, L]2 , equipped with periodic boundary conditions and
zero spatial average with initial condition u(x, t0 ) = u0 (x) for x ∈ [0, L]2 .
Observational measurements were represented by P = Ph , where Ph is the
orthogonal projection onto the Fourier modes exp(2πik · x/L) with wave
numbers k ∈ Z2 \ {0} such that 0 < |k| ≤ L/h. Here ν > 0 is the kinematic
viscosity, p(x, t) is the pressure and f (x) is a time-independent body force
with zero spatial average acting on the fluid. For simplicity, it was assumed,
as we shall here, that ∇ · f = 0.
2
The two-dimensional incompressible Navier–Stokes equations are amenable to mathematical analysis while at the same time they posses non-linear
dynamics similar to the partial differential equations that govern realistic
physical phenomenon. Using the functional notation of Constantin and Foias
[6], see also Temam [20] or Robinson [19], write (3) in the form (1) by setting
F(u) = −νAu − B(u, u) + f
(4)
where A and B are the continuous extensions of the operators given by
A = −Pσ ∆u
B(u, v) = Pσ (u · ∇v)
and
when u, v are smooth divergence-free L-periodic functions, and Pσ is the
Leray–Helmholtz projector. We recall that
Pσ (u) =
∑
k∈Z2 \{0}
{
}
k · uk
uk −
k exp(2πik · x/L)
|k|2
and also that A: V 1 → V −1 and B: V 1 × V 1 → V −1 where V α is the closure
of V, the space of zero-average R2 -valued divergence-free L-periodic trigonometric polynomials, with respect to the norm
∥u∥V α = L2
∑
|k|2α |ˆ
uk |2 .
k∈Z2 \{0}
For notational convenience we shall write V = V 1 throughout the remainder
of this paper.
Consider the data assimilation method given by (2). Using the theory of
determining modes it was shown as Theorem 1.5 in [17] that if h satisfies
L2
≥ c1 G
h2
(5)
then ∥u(t) − p(t) − q(t)∥V → 0, exponentially fast as t → ∞. Here c1
is a universal constant and G = (L/2πν)2 ∥f ∥L2 is the Grashof number.
Computations in [18] considered a fixed forcing function f = f121 scaled
to obtain different values for G. In that work the subscript 121 was used
to indicate that the force was supported on an annulus around k 2 = 121 in
Fourier space. More details on f are provided by equation (14) and Figure 1
below. For Grashof numbers between 500 000 and 60 000 000 it was shown
that the projection Ph onto the lowest 80 Fourier modes was necessary and
sufficient to ensure numerically that ∥u(t) − p(t) − q(t)∥V → 0, as t → ∞.
Since the rank of Ph scales as L2 /h2 , this is significantly less than the millions
3
suggested by the analytical bound (5). Thus, the data assimilation algorithm
given by (2) performs far better than the analysis suggests. Note, however,
that this algorithm is not suitable when the observations are given by nodal
measurements of the velocity field.
An approach to data assimilation, for dynamics governed by equations
(3) or equivalently (4), which is applicable to nodal observations, was introduced and analyzed by Azouani, Olson and Titi in [1]. Let Ih be a general
interpolant observable satisfying the approximation identity inequality
∥u − Ih (u)∥2L2 ≤ γ1 h2 ∥u∥2H 1 + γ2 h4 ∥u∥2H 2 .
(6)
Given Ih (u(t)) for t ∈ [t0 , t∗ ], solve
dv
= F(v) + µPσ (Ih (u) − Ih (v)),
v(t0 ) = v0 ,
(7)
dt
where v0 is an arbitrary initial condition. The constant µ is a relaxation
(nudging) parameter which controls the strength of the feedback control
(nudging term). In particular, the nudging term pushes the large spatial
scales of the approximating solution v toward those of the reference solution
u while the viscosity stabilizes and dissipates the fine spatial scales and any
spillover into the fine scales caused by the nudging term. It follows from
Theorem 2 equation (39) of [1] that if h and µ satisfy
L2
c0 L 2 µ
≥
≥ c2 G(1 + log(1 + G)),
(8)
h2
ν
then ∥u(t) − v(t)∥V → 0 as t → ∞. Here c2 is a universal constant and c0 is
a constant depending only on γ1 and γ2 of (6).
The number of nodal measurements needed to uniquely determine a solution to the two-dimensional Navier–Stokes equations, as t → ∞, was found
by Foias and Temam in [9] and further refined in Jones and Titi [12]. Up
to a logarithmic correction, the analytic bounds given by (8) are the same
as those given in [12]. In this paper we check the numerical performance of
the data assimilation algorithm (7) using nodal measurements given by Ih
for the same body forcing f considered in [18] scaled so that G = 2 500 000.
Let Qi be disjoint squares that cover [0, L]2 with centers xi and sides of
length h = L/K, where K 2 = N . An interpolant operator based on the nodal
measurements u(xi , t), for i = 1, 2, . . . , N , and t ∈ [t0 , t∗ ], which satisfies (6)
is
1 ∫
Ih (u)(x, t) = Ih (u)(x, t) − 2
Ih (u)(x, t) dx,
(9)
L [0,L]2
where
Ih (u)(x, t) =
N
∑
i=1
4
u(xi , t)χQi (x).
(10)
We also consider the smoothed interpolation
1 ∫
I˜h (u)(x, t) = Ieh (u)(x, t) − 2
Ieh (u)(x, t) dx,
L [0,L]2
where
Ieh (u)(x, t) =
N
∑
u(xi , t)(ρϵ ∗ χQi )(x),
(11)
(12)
i=1
and ρϵ (x) = ϵ−2 ρ(x/ϵ) with


(
1
1
+
K0 exp
2
ρ(ξ) =
1 − ξ1 1 − ξ22

0
and
K0−1
∫
=
1
−1
∫
1
)
for |ξ1 |, |ξ2 | < 1
(13)
otherwise,
)
(
1
1
+
dξ2 dξ1 .
exp
2
1 − ξ1 1 − ξ22
−1
To make the smoothing scale compatible with the resolution parameter
we take ϵ = ηh for some fixed η > 0. An analysis of a smoothed interpolant
similar to I˜h appears in Appendix A of [1] for η = 0.1 and shows that I˜h
satisfies (6). Note also that I˜h → Ih as η → 0. When η is between 0 and 1
the convolution reduces the high-frequencies that would otherwise be present
in the Fourier series representation of the characteristic function χQi . Values
of η greater than 1 blur nearby nodal measurements together. This further
downscaling could be useful in the presence of noisy measurements, see for
example [3]; however, the measurements studied here will be error free. In
this work we vary η between 0.1 and 2.0 and find that η = 0.7 leads to near
optimal performance for our data assimilation experiments.
In particular, our results show that ∥u(t) − v(t)∥V → 0, as t → ∞, when
the resolution K of the observational measurements satisfies K ≥ 8 and µ
and η are appropriately chosen. Moreover, if K ≥ 9 and η = 0.7, there is
a wide range of values for µ such that the algorithm works well. Since 64
and 81 nodes are comparable in resolution to 80 Fourier modes, the numerical efficiency of algorithm (7), using nodal measurements, is comparable to
algorithm (2) using Fourier modes [17, 18].
Rigorous mathematical analysis of the method of data assimilation studied computationally in this paper has recently been generalized to B´enard
convection by Farhat, Jolly and Titi [7] where it was shown that only observational measurements of the velocity field is sufficient to recover the full
reference solution, i.e., the velocity field and the temperature. Inspired by [7]
Farhat, Lunasin and Titi [8] have recently improved the algorithm studied
here, i.e. the one introduced in [1], by showing that it is sufficient to use
5
observational measurements of only one component of the velocity field to
recover the full reference solution. Further implementation of this algorithm,
for the subcritical surface quasi-geostrophic equation, has recently been established by Jolly, Martinez and Titi [11]. The algorithm studied here is also
closely related the 3DVAR data assimilation method developed by Bl¨omker,
Law, Stuart and Zygalakis [4] for the Navier–Stokes equations and by Law,
Shukla and Stuart [14] for Lorenz equations.
This paper is organized as follows. Section 2 describes the physical parameters, forcing and initial conditions used to generate the reference solution
to the two-dimensional incompressible Navier–Stokes equations that we will
be observing through nodal measurements of the velocity field. Section 3
reports our computational results, and section 4 gives details of our computational methods. The last section concludes that data assimilation of nodal
measurements, by means of equation (7), as studied in this paper works
computationally just as efficiently as equation (2) with Fourier modes.
2
The Reference Solution
To focus on how the smoothing and resolution of the observational measurements affect algorithm (7), as well as how to optimize the value of the
relaxation (nudging) parameter µ, we fix the viscosity and the size of the
periodic box so that
ν = 0.0001
and
L = 2π
for the remainder of this paper. We further perform all our simulations using the same reference solution u(t) to the two-dimensional incompressible
Navier–Stokes equations. As shown in [17] the spatial structure of the function f used to force the reference solution can have a significant effect on
data assimilation. Therefore, to allow comparison with previous results, we
use the exact same forcing function defined in [17], and further studied in
[18], scaled such that G = 2 500 000 in our present computations.
This function f is supported on the annulus in Fourier space with wave
numbers k such that 110 ≤ k 2 ≤ 132. In particular,
f (x) =
∑
fˆk exp(ik · x)
(14)
110≤k2 ≤132
with fˆk = fˆ−k and k · fˆk = 0, where the values of fˆk are given by Table 1
in [18] scaled to obtain the desired Grashof number. Note that this forcing
6
is at length scales of about 1/11-th the size of the periodic box. This fact is
further reflected in the level curves of
∂f2 ∂f1
curl f = curl(f1 , f2 ) =
−
∂x
∂y
depicted in Figure 1 on the left.
Figure 1: Left are contours of curl f ; right are vorticity contours of u0 .
The initial condition u0 used for our data assimilation experiments was
obtained by solving (3) with zero initial condition at time t = −25000 until
time t = 0. In terms of eddy turnover times, this ensures that more than
500 eddy turnovers have occurred before reaching t = 0. Integrating for this
length of time ensures the initial condition u0 lies close to the global attractor
and therefore reflects the energetics of the forcing f . In particular, the way
in which we initialized the solution at time t = −25000 is unimportant.
The vorticity contours ω0 = curl u0 of the initial condition u0 are depicted
in Figure 1 on the right. While the forcing f contains no Fourier modes
with wave numbers k such that |k| < 10, the initial condition u0 clearly
possesses two large eddies the size of the box. These large box-filling eddies
apparently result from the inverse cascade of energy in the two-dimensional
Navier–Stokes equations. This can be seen more clearly by examining the
energy spectrum of the reference solution.
Given a solution u(t) to the two-dimensional Navier–Stokes equations for
t ∈ [0, T ] where T = 25 000 define the average energy spectrum as
E(r) =
∑
4π 2 ∫ T ∑
|ˆ
uk (t)|2 dt where u(t) =
uˆk (t)eik·x
T 0 k∈Jr
k∈Z2 \{0}
and Jr = { k ∈ Z2 : r − 0.5 < |k| ≤ r + 0.5 }. For the reference solution
described above with initial condition u0 and time t0 = 0, the average energy
7
spectrum appears in Figure 2. While we do not see the Kraichnan scaling
of k −3 in the inertial range, we do see the Kolmogorov k −5/3 scaling in the
inverse cascade. Such spectra have been observed in other numerical experiments, see, for example, Xiao, Wan, Chen and Eyink [21]. In particular, the
inverse cascade appears responsible for the box-filling eddies observed in the
initial condition u0 which persist for all times t > 0.
Figure 2: Time averaged energy spectrum of the reference solution.
1
E(r)
forcing
r^(-5/3)
r^(-5)
0.01
inverse cascade
energy
0.0001
1e-06
inertial
range
1e-08
1e-10
1e-12
dissipation
range
1e-14
1
10
100
r
We compute the eddy turnover time for the reference solution as
τ = 4π 2
∞
∑
r−1 E(r)/
∞
(∑
r=1
)3/2
E(r)
≈ 30.8
r=1
and conclude that our averages have been computed over T /τ ≈ 812 eddy
turnovers. The spectrum of τ f , which also has units of energy, has been
plotted in Figure 2 as three circles to illustrate where the forcing lies in
relation to the energy spectrum. Note that the forcing exactly divides the
energy spectrum between the part which scales as k −5/3 and the part which
scales as k −5 .
Having, to some extent, described the reference solution that will be used
in our numerical experiments, we now turn to our main point of study, the
data assimilation of nodal measurements of the velocity field.
8
3
Nodal Observations of Velocity
We consider nodal observations u(xi , t), for i = 1, . . . , N , of the reference solution u, that was computed according to the incompressible two-dimensional
Navier–Stokes equations (3) and initialized with u0 as described in Figure 1,
and interpolate these measurements using the operator Ih defined by (9).
The resulting equations for the approximating solution v may be written as
dv
+ νAv + B(v, v) = f − µPσ (Ih (u) − Ih (v))
dt
(15)
where v is initialized as v0 = 0. Note that only the observations Ih (u) of
the reference solution u enter into the equations for computing for v. Also
note that ∥u(0) − v(0)∥V = ∥u0 ∥V ≈ 1.946. Our goal now is to choose the
resolution parameter h and the relaxation (nudging) parameter µ in such a
way that ∥u(t) − v(t)∥V → 0, numerically, as t → ∞.
As discussed in [1], if µ is too small, the feedback control (nudging term)
will be too weak to ensure the approximating solution converges to the reference solution. If µ is too large, then spill over into the fine scales becomes
significant and again prevents recovery of the reference solution. Figure 3 illustrates each of these possibilities for h = L/K, where K = 9, using different
values of µ.
Figure 3: The error ∥u(t) − v(t)∥V versus t for h = 0.6981.
µ=1/2
µ=1/5
µ=2/9
µ=1/4
µ=1/3
100
1
error
0.01
0.0001
1e-06
1e-08
1e-10
1e-12
1e-14
0
5000
10000
15000
time
20000
25000
When µ = 1/2 the relaxation (nudging) parameter is too large for the
approximating solution to converge to the reference solution, and when µ =
9
1/5 it is too small. However, the intermediate value µ = 1/3 works with the
error represented by ∥u(t) − v(t)∥V falling below 10−10 by t = 13 417.8. Note
that, since the double-precision floating-point numbers used to represent the
Fourier modes of u and v on the computer have only 15 digits of precision,
we don’t expect convergence of ∥u(t) − v(t)∥V to exact zero over time.
For µ = 1/4 the error falls below 10−10 at T = 12 327.1, however, it
rises again and it is not clear whether after T = 23 463.9 the error finally
stays below 10−10 or not. The value µ = 2/9 shows an even more irregular
pattern where ∥u(t) − v(t)∥V exhibits a period of decay followed by a period
of growth that covers six orders of magnitude. Fortunately, most of our
parameter choices avoid these borderline cases and the corresponding error
either converges towards zero and stays below 10−10 or shows few signs of
converging and stays well above 10−10 .
To determine the values of h and µ for which it is possible to recover the
reference solution to within numerical roundoff error we fix ϵ > 0 and define
Tmax = sup { t ∈ [0, T ] : ∥v(t) − u(t)∥V ≥ ϵ }
and
Tmin = inf { t ∈ [0, T ] : ∥v(t) − u(t)∥V ≤ ϵ }
Let Tmax = 0, when the supremum is over the empty set, and Tmin = ∞ when
the infimum is over the empty set. When ∥v(T ) − u(T )∥V ≥ ϵ further set
Tmax = ∞ to ensure Tmax ≥ Tmin . Inspired by Figure 3 we also define
εavg
1 ∫T
=
∥v(t) − u(t)∥V dt
T − T0 T0
and take ϵ = 10−10 , T = 25 000 and T0 = 2T /3 for our numerics.
Table 1 shows the results of our computational experiments. Runs with
K = 8 were also performed, however, no value of µ yielded a finite value for
Tmax or Tmin or even an approximation for which the error ∥u(t) − v(t)∥V fell
below 10−2 . We conclude that K = 9 is the minimal resolution for which
there exists a µ such that the error tends toward zero. At this minimal
resolution only a narrow range of values for µ near 1/3, result in an error
which falls below 10−10 . When K = 10, there is a much greater range of
corresponding values for µ that work well. In fact, when K = 10 all values of
µ between 1/6 and 1/2 led to corresponding approximations v(t) such that
εavg ≈ 3 × 10−14 . Since the double-precision floating-point numbers used to
represent the Fourier modes of u and v on the computer have only 15 digits of
precision, the fact that the error can approach 10−14 is remarkable. We again
note, as is consistent with the analysis in [1], our numerical experiments do
not recover the reference solution if µ is too small or too large.
10
Table 1: Data assimilation using Ih .
K=9
Tmax
∞
∞
∞
∞
∞
∞
∞
23463.9
13466.2
13417.8
∞
∞
∞
∞
µ
Tmin
0.0625
∞
0.125
∞
0.154
∞
0.167
∞
0.182
∞
∞
0.2
0.222
∞
0.25 12327.1
0.286 12275.6
0.333 13417.8
0.4
∞
∞
0.5
0.6
∞
0.7
∞
εavg
1.1
3.9 × 10−1
2.4 × 10−1
2.4 × 10−1
1.5 × 10−1
3.5 × 10−2
1.9 × 10−5
3.7 × 10−10
1.3 × 10−12
1.4 × 10−12
1.3 × 10−1
3.2 × 10−1
6.7 × 10−1
1.5
Tmin
∞
13754.5
4331.1
3965.0
3320.1
2825.2
2870.1
2701.0
2581.4
2601.8
3008.4
4564.3
8604.5
∞
K = 10
Tmax εavg
∞ 8.8 × 10−1
17094.3 4.1 × 10−12
4432.6 5.4 × 10−14
4187.7 3.1 × 10−14
3320.1 2.8 × 10−14
2825.2 2.8 × 10−14
2870.1 2.5 × 10−14
2701.0 2.5 × 10−14
2598.6 2.1 × 10−14
2601.8 2.0 × 10−14
3008.4 2.1 × 10−14
4564.3 2.4 × 10−14
9964.6 5.7 × 10−14
∞ 5.3 × 10−2
Figure 4: Tmax versus µ for ϵ = 10−10 , T = 25000 and K = 10.
6000
5000
Tmax
4000
3000
2000
η=0
η=1
η=0.9
η=0.7
η=0.5
1000
0
0.1
1
10
100
µ
Next consider the family of smoothed interpolants I˜h for different values of
η. Figure 4 plots Tmax versus µ. When η is near 0.7 we find that values of µ between 1/4 and 64 all lead to approximations such that ∥u(t)−v(t)∥V ≤ 10−10
for large enough T . Thus, smoothing with η near 0.7 leads to a significantly
wider range of values for µ such that the data assimilation algorithm can be
11
used to recover the reference solution. Note when η = 2 and K = 10 that
no values of µ led to the convergence of the approximating solution to the
reference solution over time.
Having found good values for η we continue our numerical study by fixing
η = 0.7 and varying µ for different resolutions h = L/K, where K = 8, 9
and 10. As before ϵ = 10−10 , T = 25 000 and T0 = 2T /3. The computational results given in Table 2 show that observational measurements with a
resolution given by K = 8 can now lead to an approximate solution which
converges to the reference solution over time. Moreover, the accuracy of the
resulting approximations also improve compared to the non-smoothed case.
Note that we have omitted reporting Tmin in Table 2 since in all cases Tmin
was equal or nearly equal to Tmax .
Table 2: Data assimilation using I˜h where η = 0.7.
µ
0.25
0.5
1
2
4
8
16
32
64
128
K=8
Tmax εavg
∞ 5.1 × 10−1
∞ 6.5 × 10−2
2817.8 1.1 × 10−13
2527.1 2.6 × 10−14
2013.1 2.1 × 10−14
2191.6 2.1 × 10−14
4137.3 4.7 × 10−14
∞ 3.0 × 10−1
∞ 1.4
∞ 2.6
K=9
Tmax εavg
∞ 9.9 × 10−2
2570.1 2.4 × 10−14
1686.5 1.8 × 10−14
1232.6 1.7 × 10−14
1092.4 1.7 × 10−14
1124.8 1.7 × 10−14
1360.4 1.7 × 10−14
2752.9 2.1 × 10−14
∞ 4.9 × 10−1
∞ 1.9
K = 10
Tmax εavg
2834.2 2.4 × 10−14
1534.2 1.7 × 10−14
1112.7 1.6 × 10−14
897.5 1.6 × 10−14
718.9 1.6 × 10−14
717.8 1.6 × 10−14
769.7 1.6 × 10−14
1284.2 1.7 × 10−14
2848.6 2.1 × 10−14
∞ 1.1
Figure 5 plots the data in Table 2. From this figure it is clear that the
data assimilation algorithm given by (15) with Ih replaced by I˜h works well
when η = 0.7 for a wide range of values of the relaxation parameter µ.
4
Numerical Methods
All fluid dynamics simulations presented in this paper were performed using
a new parallel code written for the NVIDIA Compute Unified Device Architecture in the CUDA C programming language [16] which was developed on
desktops at the University of Nevada Reno and run on the Big Red II Cray
XE6/XK7 supercomputer at Indiana University. Computations were made
in the stream function formulation using a fully-dealiased spectral Galerkin
12
Figure 5: Tmax versus µ for ϵ = 10−10 , T = 25000 and η = 0.7.
6000
5000
Tmax
4000
3000
2000
1000
K=8
K=9
K=10
0
0.1
1
10
100
µ
method. Time steps were performed using a split-Euler method in which the
linear term was integrated exactly and the non-linear terms were integrated
using forward differences.
Specifically we set ∆Ψ = curl u and compute the reference solution using
the stream function formulation
∂∆Ψ
− ν∆2 Ψ + β(Ψ) = curl f,
(16)
∂t
where
β(Ψ) = J(Ψ, ∆Ψ) = Ψx ∆Ψy − Ψy ∆Ψx
= ((Ψx )2 − (Ψy )2 )xy − (Ψx Ψy )xx + (Ψx Ψy )yy .
Similarly, set ∆Φ = curl v and compute the approximating solution using
∂∆Φ
− ν∆2 Φ + β(Φ) = curl f − µ(Rh (Φ) − Rh (Ψ)),
∂t
(17)
where Rh (Ψ) = curl Pσ Ih (curl−1 ∆Ψ). Note that Rh : V 3 → V −1 .
Following the 2/3 dealiasing rule applied to 512×512 sized discrete Fourier
transforms we set K = {−341, . . . , 341}2 and approximate
Ψ≈
∑
ˆ k eik·x ,
Ψ
Φ≈
∑
ˆ k eik·x
Φ
and
curl f =
gˆk eik·x .
k∈K
k∈K
k∈K
∑
Substituting these approximations into (16) and (17) and projecting onto the
Fourier modes with wave numbers k ∈ K yields the Galerkin truncations
−
ˆk
dΨ
ˆ
ˆ k k 4 + β(Ψ)
k2 − ν Ψ
ˆk
k = g
dt
13
and
ˆk
dΦ
ˆ k = gˆk − µR
ˆ k k 4 + β(Φ)
ˆ h (Φ − Ψ)k .
k2 − ν Φ
dt
The corresponding numerical scheme for the reference solution is
−
{
}
1
2
ˆ
ˆ k (t + ∆t) ≈ e−νk2 ∆t Ψ
ˆ k (t) + ∆t β(Ψ(t))
Ψ
− 4 gˆk (1 − e−νk ∆t )
k
2
k
νk
and for the approximating solution is
{
(
)}
ˆ
ˆ k (t + ∆t) ≈ e−νk2 ∆t Φ
ˆ k (t) + ∆t β(Φ(t))
ˆ h (Φ − Ψ)k
Φ
+
µ
R
k
k2
1
2
− 4 gˆk (1 − e−νk ∆t ).
νk
At the discrete level it is still the case that only nodal-point observational
measurements of the reference solution are used to construct the approximating solution. Moreover, since both solutions are integrated using the same
numerical methods, we may think of Ψ as an unknown reference solution
that evolves according to a known discrete dynamical system, and the solution represented by Φ as an approximation generated by data assimilation
according to the exact same discrete dynamics. Thus, even though our numerical schemes are only first order in time, we consider our numerical experiments to simulate data assimilation in the absence of both measurement
and model errors.
We take the time step to be ∆t = 1/2048 which is enough to ensure that
the CFL condition
N ∆t
sup {|u1 (x)| + |u2 (x)|} ≤ 0.1608 ≪ 1
2L x∈Ω
is satisfied for the reference solution over the entire run. Note that as µ
gets larger the data assimilation equations (15) become stiffer. Therefore,
the time step ∆t was also chosen small enough to ensure the stability of the
coupled numerical scheme for computing the approximating solution.
Our numerical software has been optimized so that it runs entirely on the
CUDA hardware with zero memory copies and four fast Fourier transforms
per time step. Note that four transforms is the minimal number for the twodimensional Navier–Stokes equations, see Basdevant [2] for further remarks
and analogous optimizations when computing the three-dimensional Navier–
Stokes equations. As device memory on the CUDA hardware is relatively
scarce we also minimized our storage requirements. Storage requirements
consist of four n × n double-precision scalar arrays: one for Ψ, another for
14
Figure 6: Data Flow for Computing the Non-linear Term
T1 :
T2 :
ˆ xIFFT / Ψx
/ Ψ2 − Ψ2
Ψ
x
y
FF
FF xx;
FFxx
xx FFF
F#
xx
x
IFFT /
ˆ
/
Ψy
Ψy
Ψx Ψy
FFT
FFT
2
/ Ψ2 d
x − Ψy
HH
HH
HH
HH
H$
ˆ
/ Ψd
/ β(Ψ)
x Ψy
Φ and two temporary arrays T1 and T2 . Figure 6 shows the data flow when
computing the non-linear term. The first line represent the contents of T1 ,
the second represents T2 and the arrows represent computational kernels.
When using 512 × 512 FFTs, our code achieves approximately 1062 time
steps per second. In particular, the computational speed running on an
NVIDIA Tesla K20 GPU was found to be roughly 37 times faster than equivalent CPU code running on single AMD Opteron 6212 core and 11.5 times
faster when compared to running on 32 CPU cores. Correctness of operation
was verified using the Navier–Stokes solver described in [17] and [18].
5
Conclusion
As is consistent with the analytical bound (8) and related discussion in [1], the
numerical results given in Table 1 and Figure 5 show that the approximating
solution does not converge to the reference solution when µ is either too small
or too big. At the same time, provided the resolution h is fine enough and
η ≈ 0.7, there is a wide range of good values for µ when using the smoothed
interpolant observable I˜h . In particular, when h = L/10 ≈ 0.6283, the data
assimilation algorithm (7) performs similarly for values of µ between 0.5 and
32. Note, however, that smaller values of µ are computationally preferable
because of stiffness considerations.
We remark that our numerical experiments have been conducted using
exact error-free measurements and exact model dynamics and that in the
presence of measurement and model errors we don’t expect a similarly wide
range of good values for µ. In fact, preliminary computations show when is
noise added to the system that there exists a unique optimal value for µ reflecting the tradeoff between measurement and model errors. Theoretically, if
the dynamics represented by F(u) in (1) are linear, then µ can be seen as the
parameter of a linear Kalman filter, see for example Majda and Harlim [15],
and there exists an analytically derived optimal value for µ which represents
the tradeoff between measurement and model errors. In the fully non-linear
case studied here, the fact that there is a wide range of good constant values
for µ in the absence of measurement and model errors suggests, provided the
15
resolution h is fine enough, that µ can be further optimized as if the model
was linear.
We now compare the coarsest resolution h = L/K that works for the
data assimilation experiments presented here when η = 0.7 to the number
of numerically determining modes found in [18]. Under the same physical
parameters and forcing, the minimum number of Fourier modes needed by
(2) was
nc = card(D5 ) = 80
where
DR = { eik·x : 0 < k12 + k22 ≤ R2 }.
In this paper we show the minimum of nodal measurements needed are
N = K 2 = 64
which by the Nyquist–Shannon sampling theorem may be represented by the
Fourier modes
NK = { eik·x : 0 < max(|k1 |, |k2 |) ≤ K/2 }.
To compare these two results we note that DR represents a circle in Fourier
space while NK represents a square. Let Rmin = 5 and Kmin = 8. If the
resolution requirements of algorithm (2) are comparable to (7), we would
expect that NKmin ⊆ DR would imply R ≥ Rmin and that DRmin ⊆ NK would
imply K ≥ Kmin . This is supported by our results. If N8 ⊆ DR then
√
8 2
≈ 5.65 ≥ 5 = Rmin .
R≥
2
Similarly, if D5 ⊆ NK then
K ≥ 2 · 5 = 10 ≥ 8 = Kmin .
Thus, even though our nodal observations possess the problems of aliasing
and high-frequency spill over, these problems can be mediated with appropriate smoothing. The resulting resolution K needed for the approximating
solution to converge to the exact solution is then about the same as suggested
by the number of numerically determining modes.
Acknowledgements
The authors would like to thank Professor Michael Jolly for his help with
the Big Red II supercomputer at Indiana and his current collaboration on
16
research treating measurement and model error. The work of E.O. was supported in part by EPSRC grant EP/G007470/1, by sabbatical leave from the
University of Nevada Reno and by NSF grant DMS-1418928. The work of
E.S.T. was supported in part by a grant of the ONR, and the NSF grants
DMS-1109640 and DMS-1109645.
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